Download Rearranging Formulae Solving Equations Involving Fractions

solving equations
solving equations
Solving Equations Involving Fractions
MPM1D: Principles of Mathematics
Recap
Solve 56 (x − 4) =
3
10 (2x
− 1) algebraically.
The LCM of 6 and 10 is 30, so multiply both sides by 30,
distribute, and solve.
Rearranging Formulae
30 · 56 (x − 4) = 30 ·
J. Garvin
3
10 (2x
25(x − 4) = 9(2x − 1)
− 1)
25x − 100 = 18x − 9
25x − 18x = 100 − 9
7x = 91
x = 13
J. Garvin — Rearranging Formulae
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solving equations
solving equations
Rearranging Formulae
Rearranging Formulae
Consider the formula for the area of a rectangle, A = ` · w .
This new formula allows us to calculate the length directly.
If we want to determine the area for a given length and
width, we simply substitute those values into ` and w and
multiply them together.
There are many situations in which we need to rearrange
formulae, depending on which value is desired (and which
values are given).
If we want to know the value of one of the dimensions,
however, we need to rearrange the formula, isolating the
desired variable.
We can do this using the same algebraic techniques that we
used for solving equations.
For instance, if we want to calculate a rectangle’s length,
given its width and area, we can isolate ` by dividing both
sides by w .
A
` ·
w
=
w
w
A
=`
w
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J. Garvin — Rearranging Formulae
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solving equations
solving equations
Rearranging Formulae
Rearranging Formulae
Example
Example
Ohm’s Law relates the voltage, V (volts), in a conductor to
both the current, I (amps), and the resistance, R (ohms),
according to the relationship V = IR. If the voltage is 12 V,
and the resistance is 1.5 Ω, what is the current?
The formula to convert a temperature in Celsius, C
(degrees), to Fahrenheit, F (degrees), is F = 95 C + 32. Use
this formula to convert a temperature of 77◦ F to Celsius.
First, isolate C .
Divide both sides of the equation by R to isolate I .
IR
V
=
R
R
V
=I
R
Substitute the values 12 and 1.5 into the equation.
12
I =
1.5
I =8A
J. Garvin — Rearranging Formulae
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F − 32 = 95 C
5(F − 32) = 9C
5
9 (F
Now, substitute 77 into F .
− 32) = C
C = 59 (77 − 32)
C = 59 (45)
C = 25◦ C
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solving equations
solving equations
Rearranging Formulae
Rearranging Formulae
Example
Example
The volume of a cylinder is given by V = πr 2 h, where r is
the radius and h is the height. Determine the radius of a
cylinder with a height of 14 cm and a volume of 320 cm3 .
A simple pendulum of given length, ` metres,
r has a period,
`
T seconds, given by the equation T = 2π
, where g is
g
the acceleration due to gravity (typically 9.8 m/s2 ). What is
the length of a pendulum that has a period of 3 seconds?
Begin by isolating r 2 , then take the square root to isolate r .
V
= r2
πh
r
V
=r
πh
Now substitute 14 and 320 into h and V respectively.
r
320
r=
14π
r ≈ 2.7 cm
J. Garvin — Rearranging Formulae
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To isolate `, start by dividing both sides by 2π.
q
`
2π
g
T
=
2π
2π
J. Garvin — Rearranging Formulae
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solving equations
Rearranging Formulae
solving equations
Questions?
Square both sides to cancel out the square root.
2
T
`
=
2π
g
Multiply both sides by g to isolate `.
2
T
g ·`
g·
=
2π
g
2
T
g·
=`
2π
Now substitute g = 9.8 and T = 3 into the formula.
2
3
` = 9.8 ·
2π
` ≈ 2.23 m
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J. Garvin — Rearranging Formulae
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